Partitioned dual Maclaurin symmetric mean operators based on picture fuzzy sets and their applications in multi-attribute decision-making problems

The partitioned Dual Maclaurin symmetric mean (PDMSM) operator has the supremacy that can justify the interrelationship of distinct characteristics and there are a lot of exploration consequences for it. However, it has not been employed to manage “multi-attribute decision-making” (MADM) problems represented by picture fuzzy numbers. The basic inspiration of this identification is to develop the novel theory of picture fuzzy PDMSM operator, and weighted picture fuzzy PDMSM operator and to identify their important results (Idempotency, Monotonicity, and Boundedness). Further, to identify the best decision, every expert realized that they needed the best way to find the beneficial optimal using the proper decision-making procedure, for this, we diagnosed the MADM tool in the consideration of deliberated approaches based on PF information. Finally, to drive the characteristics of the invented work, several examples are utilized to test the manifest of the comparative analysis with various more existing theories, which is a fascinating and meaningful technique to deeply explain the features and exhibited of the proposed approaches.


Partitioned dual Maclaurin symmetric mean operators based on picture fuzzy sets and their applications in multi-attribute decision-making problems
Tahir Mahmood 1* , Ubaid ur Rehman 1 , Walid Emam 2 , Zeeshan Ali 3 & Haolun Wang 4 The partitioned Dual Maclaurin symmetric mean (PDMSM) operator has the supremacy that can justify the interrelationship of distinct characteristics and there are a lot of exploration consequences for it.However, it has not been employed to manage "multi-attribute decision-making" (MADM) problems represented by picture fuzzy numbers.The basic inspiration of this identification is to develop the novel theory of picture fuzzy PDMSM operator, and weighted picture fuzzy PDMSM operator and to identify their important results (Idempotency, Monotonicity, and Boundedness).Further, to identify the best decision, every expert realized that they needed the best way to find the beneficial optimal using the proper decision-making procedure, for this, we diagnosed the MADM tool in the consideration of deliberated approaches based on PF information.Finally, to drive the characteristics of the invented work, several examples are utilized to test the manifest of the comparative analysis with various more existing theories, which is a fascinating and meaningful technique to deeply explain the features and exhibited of the proposed approaches.
The aim of the decision-making (DM) sciences is to identify the massive dominant decision from a group of expected ones.MADM is a critical and crucial component of these sciences.In genuine DM strategy, the dilemma requires resolving the provided decisions by many classes like single or interval evaluation investigations.However, in many awkward scenarios, it is very challenging for the expert to reduce their decisions to a classical number.To establish it, Zadeh 1 exposed the mathematical framework of fuzzy set (FS) by proposing a new function, called membership degree (MD) defined from universal set X to [0, 1] .If we are taking any arbitrary element x ∈ X and assigning it to a value , then the resultant value should belong to [0, 1] .In many cases, the meaningful theory of FS has been unsuitable, when an expert copes with information that contains yes or no.Then for this, one of the most suitable and meaningful theories which easily manage awkward and complicated data arising in real life was invented by Atanassov 2,3 , called intuitionistic FS (IFS) by including a new function, called non-membership degree (NMD) η Y • defined from universal set X to [0, 1] .If we are taking any arbitrary element x ∈ X and assigning it to a function η Y • (x) , then the resultant value should belong to [0, 1] with 0 ≤ ζ Y • (x) + η Y • (x) ≤ 1 .Unrelatedly, the informative idea of IFSs has been utilized in various areas.Still, there is a diversity of circumstances in which IFS theory can't be employed.For instance, casting a vote and managing dilemmas like yes, abstinence, no, and refusal, which bounded the implementation of IFS theory.Then for this, one of the most suitable and meaningful theories which easily manage difficult and intricate data occurring in real life was invented by Cuong 4 , called picture FS (PFS) by including a new function, called abstinence degree (AD) ℘ Y • defined from universal set X to [0, 1] .If we are taking any arbitrary element x ∈ X and assigning it to a function ℘ Y • (x) , then the resultant value should belong to

Motivation
The theory of MSM was deduced by Maclaurin 44 and then modified by DeTemple and Roberston 45 .The MSM and dual MSM (DMSM) differ from typical aggregation operators in that they take into account the interactions between many input parameters.Because of this, the MSM and DMSM excel at delivering flexible and reliable data combinations, making it especially useful for dealing with (MADM) scenarios where the characteristics are distinct from one another.Further, to organize and optimize data storage and retrieval, a collection must be partitioned.Combining similar information, cutting down on duplication, and enhancing query efficiency, enables effective data management.Partitioning can also boost parallel processing capacities, resulting in quicker data processing and better system scalability.From the above-mentioned discussion, we observed that there are no AOs under the setting of PF information which is based on DMSM and can cope with the partitioned collection.Thus, in this manuscript, we will try to utilize PDMSM operators in the setting of PF information.The PDMSM operators in the environment of PF information have the supremacy that can justification of the interrelationship of distinct characteristics and there are a lot of exploration consequences for it.Moreover, there is no such MADM approach in the literature under PF information that can employ the PDMSM operator and cope with MADM dilemmas.So further, in this manuscript, we devise a MADM method under PF information relying on the invented PDMSM operators.The basic inspiration for this identification is described below: (1) To develop the novel theory of PFPDMSM operator, WPFPDMSM operator and identified their important results (Idempotency, Monotonicity, Boundedness).(2) To identify the best decision, every expert realized that they needed the best way to find the beneficial optimal using the proper decision-making procedure, for this, we diagnosed the MADM tool in the consideration of deliberated approaches based on PF information.(3) To drive the characteristics of the invented work, several examples are utilized to test the manifest of the comparative analysis with various more existing theories, which is a fascinating and meaningful technique to deeply explain the features and exhibited of the proposed approaches.

Organization of the paper
Various and main important of this analysis are organized the shape: In Sect."Preliminaries", we revised the concept of PF data along with their operational laws.Further, we also recalled the theory of Maclaurin symmetric mean (MSM), dual MSM (DMSM), and PDMSM operators and their related results which are very helpful for the invented work.In Sect."PDMSM Operators Based on PFNs", we established the novel theory of PFP-DMSM operator, and WPFPDMSM operator and identified their important results (Idempotency, Monotonicity, Boundedness).In Sect."Application ("MADM Process")", we identified the best decision, every expert realized that they needed the best way to find the beneficial optimal using the proper decision-making procedure, for this, we diagnosed the MADM tool in the consideration of deliberated approaches based on PF information.Finally, to drive the characteristics of the invented work, several examples are utilized to test the manifest of the comparative analysis with various more existing theories, which is a fascinating and meaningful technique to deeply explain the features and exhibited of the proposed approaches.In Sect."Conclusion", we concluded some remarks.

Preliminaries
The major impact of this scenario is to revise the theory of PF set and its operational rules.Further, we also recalled the theory of Maclaurin symmetric mean (MSM), dual MSM (DMSM), and PDMSM operators and their related results which are very helpful for the invented work.
Definition 1: 4 A PFS M on X is deliberated by: , η 2 be two PFNs, ℧ be a positive real number, then Moreover, using Eqs.( 2)-( 5), we obtain the following theories, such that. ( ℧ > 0 Additionally, finding the ranking between any PFNs is a very challenging task for experts, for this, we revise the score value (SV) and accuracy value (AV), such that Then by employing Eqs. ( 6) and ( 7), we diagnose.
Proposition 1: 22 when the parameter F is fixed.

PDMSM operators based on PFNs
PDMSM operator has the supremacy which can take justification of interrelationship of distinct characteristics and there are a lot of exploration consequences on it.However, it has not been employed to manage MADM dilemmas represented by PFNs.The basic inspiration for this identification is to develop the novel theory of the PFPDMSM operator, and WPFPDMSM operator and identify their important results (Idempotency, Monotonicity, Boundedness).

Definition 4: Let a gathering of PFNs
Then the underneath expression.demonstrate PFPDMSM operators.Where ć reveals number of partitions P J 1, P J 2 , . . ., P J ć , F = 1, 2, ...., is a param- eter, ς r is the amount of attributes in P r ,(t 1 , t 2 , ..., t F ) covers all the b-tuple combination of (1, 2, .., ς r ) and Theorem 1: Let a gathering of PFNs M t = ζ t , ℘ t , η t (t = 1, 2, . . ., f) .Then by employing Eq. ( 14), we get.To develop AOs that adhere to specific logical and mathematical principles, monotonicity, boundedness, and idempotency serve as a foundation.This improves the usefulness and interpretability of the aggregated data.For aggregation operators to guarantee the dependability and significance of the aggregation process, the properties of monotonicity, boundedness, and idempotency are essential.Monotonicity preserves the intuitive idea that more substantial contributions should result in a bigger aggregate by ensuring that as input values grow, the aggregated output does not decrease.Extreme outliers are prevented from unduly affecting the conclusion by boundedness, which ensures that the aggregate result stays within a certain range.Idempotency guarantees that the same outcome is produced when the AOs are applied again to the same set of data, improving the stability and predictability of the aggregate process.Together, these characteristics provide consistency, precision, and control over the aggregation process, which qualifies it for a variety of applications including statistical analysis, decision-making, and data summarization.

Proposition 6: (Monotonicity) Let two gatherings of PFNs
when F is fixed.
Proof: Omitted.Additionally, we diagnose some properties, such that.

Proposition 8: (Commutativity) Let a gathering of PFNs
when F is fixed.Further, if we choose the value of ć = 1, then we get

Application ("MADM process")
MADM is a crucial and essential part of the decision-making sciences whose aim is to recognize the massive dominant decision from the group of expected ones.In a genuine decision-making strategy, the problem needs to resolve the provided decisions by many classes like single or interval evaluation investigations.However, in many awkward scenarios, it is very challenging for the expert to reduce their decisions to a classical number.However, it has not been employed to manage MADM dilemmas represented by PFNs.The basic inspiration for this identification was to develop the novel theory of PFPDMSM operator, and WPFPDMSM operator and identify their important results (Idempotency, Monotonicity, Boundedness).The primary goal of this section is to identify the best decision, every expert realized that they needed the best way to find the beneficial optimal using the proper DM procedure, for this, we diagnosed the MADM tool in the consideration of deliberated approaches based on PF information.Finally, to drive the characteristics of the invented work, several examples are utilized to test the manifest of the comparative analysis with numerous more prevailing theories, which is a fascinating and meaningful technique to deeply explain the features and exhibited of the proposed approaches.

Decision-making technique
In this scenario, we aim to diagnose a new MADM process based on the environment for solving DM dilemmas.For this, we assume T = T 1 , T 2 , ..., T n , and = { 1 , 2 , ..., F } be a group of alternatives and attributes with ω , ..., T n is partitioned into ć distinct groups P J 1 , P J 2 , ..., P J ć .Therefore, to identify the best decision, every expert realized that they needed the best way to find the beneficial optimal using the proper decision-making procedure.For this, we computed the procedure, whose main steps are defined below.
Step 1 In DM dilemmas there are two various sort of attributes, that is cost sort and benefit sort, to abolish the effect of different attribute types, transform the decision matrix , 1 ≤ t ≤ n, 1 ≤ y ≤ f .To normalize the decision matrix use the given formula: where 1 ≤ t ≤ n, 1 ≤ y ≤ f. Step

Numerical example
Assume the five suitable enterprise resource management (ERP) schemes expressed by {T 1 , T 2 , T 3 , T 4 , T 5 } denoted the collection alternatives, which require to be resolved by decision-makers.For this, ERP schemes are resolved based on the four attributes expressed by { 1 , 2 , 3 , 4 } , represented by: 1 : Technical Achievement; 2 : Human Recourse; 3 : Economic benefits and 4 : constructions of the enterprises.For every attribute, the expert gives their opinion in the shape of a weight vector like 0.4, 0.3, 0.2, and 0.1.Therefore, the selection process of the ERP scheme is evaluated with the help ofthe above procedure.
To normalize the decision matrix use the given formula: where 1 ≤ t ≤ n, 1 ≤ Y ≤ f .But the information given in Table 1, does not required to be normalized.
Step 4 Make a comparison between S M ′ 1 , S M ′ 2 , ...., S M ′ n and Ǎ M ′ 1 , Ǎ M ′ 2 , ..., Ǎ M ′ n , then ranking all the alternatives to choose the best one, such that From the above, we get the best optimal in the shape of T 2 .

Comparative analysis
Comparative analysis is an old technique, used for comparing two or more theories.As a main theme of this analysis, we described the advantages and disadvantages of the invented work with the help of comparison among any two or more theories.For this, we suggested various existing theories, described as.
• AOs for IF information, deduced by Xu 18 .

2
Using the developed WPFPDMSM operator to get Noted that M ′ t is the preference argument of alternative T t , t = 1, 2, ..., n Step 3 Obtained values S M ′ t and Ǎ M ′ t of aggregated outcomes M ′ t (t = 1, 2, ..., n).